using System;
using System.Runtime.InteropServices;
using System.Runtime.Serialization;

namespace DotNetMatrix
{
	
	/// <summary>Eigenvalues and eigenvectors of a real matrix. 
	/// If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is
	/// diagonal and the eigenvector matrix V is orthogonal.
	/// I.e. A = V.Multiply(D.Multiply(V.Transpose())) and 
	/// V.Multiply(V.Transpose()) equals the identity matrix.
	/// If A is not symmetric, then the eigenvalue matrix D is block diagonal
	/// with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues,
	/// lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda].  The
	/// columns of V represent the eigenvectors in the sense that A*V = V*D,
	/// i.e. A.Multiply(V) equals V.Multiply(D).  The matrix V may be badly
	/// conditioned, or even singular, so the validity of the equation
	/// A = V*D*Inverse(V) depends upon V.cond().
	/// 
	/// </summary>
	
	[Serializable]
	public class EigenvalueDecomposition : System.Runtime.Serialization.ISerializable
	{
		#region	 Class variables
		
		/// <summary>Row and column dimension (square matrix).
		/// @serial matrix dimension.
		/// </summary>
		private int n;
		
		/// <summary>Symmetry flag.
		/// @serial internal symmetry flag.
		/// </summary>
		private bool issymmetric;
		
		/// <summary>Arrays for internal storage of eigenvalues.
		/// @serial internal storage of eigenvalues.
		/// </summary>
		private double[] d, e;
		
		/// <summary>Array for internal storage of eigenvectors.
		/// @serial internal storage of eigenvectors.
		/// </summary>
		private double[][] V;
		
		/// <summary>Array for internal storage of nonsymmetric Hessenberg form.
		/// @serial internal storage of nonsymmetric Hessenberg form.
		/// </summary>
		private double[][] H;
		
		/// <summary>Working storage for nonsymmetric algorithm.
		/// @serial working storage for nonsymmetric algorithm.
		/// </summary>
		private double[] ort;
		
		#endregion //  Class variables

		#region Private Methods
		
		// Symmetric Householder reduction to tridiagonal form.
		
		private void  tred2()
		{
			//  This is derived from the Algol procedures tred2 by
			//  Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
			//  Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
			//  Fortran subroutine in EISPACK.
			
			for (int j = 0; j < n; j++)
			{
				d[j] = V[n - 1][j];
			}
			
			// Householder reduction to tridiagonal form.
			
			for (int i = n - 1; i > 0; i--)
			{
				// Scale to avoid under/overflow.
				
				double scale = 0.0;
				double h = 0.0;
				for (int k = 0; k < i; k++)
				{
					scale = scale + System.Math.Abs(d[k]);
				}
				if (scale == 0.0)
				{
					e[i] = d[i - 1];
					for (int j = 0; j < i; j++)
					{
						d[j] = V[i - 1][j];
						V[i][j] = 0.0;
						V[j][i] = 0.0;
					}
				}
				else
				{
					// Generate Householder vector.
					
					for (int k = 0; k < i; k++)
					{
						d[k] /= scale;
						h += d[k] * d[k];
					}
					double f = d[i - 1];
					double g = System.Math.Sqrt(h);
					if (f > 0)
					{
						g = - g;
					}
					e[i] = scale * g;
					h = h - f * g;
					d[i - 1] = f - g;
					for (int j = 0; j < i; j++)
					{
						e[j] = 0.0;
					}
					
					// Apply similarity transformation to remaining columns.
					
					for (int j = 0; j < i; j++)
					{
						f = d[j];
						V[j][i] = f;
						g = e[j] + V[j][j] * f;
						for (int k = j + 1; k <= i - 1; k++)
						{
							g += V[k][j] * d[k];
							e[k] += V[k][j] * f;
						}
						e[j] = g;
					}
					f = 0.0;
					for (int j = 0; j < i; j++)
					{
						e[j] /= h;
						f += e[j] * d[j];
					}
					double hh = f / (h + h);
					for (int j = 0; j < i; j++)
					{
						e[j] -= hh * d[j];
					}
					for (int j = 0; j < i; j++)
					{
						f = d[j];
						g = e[j];
						for (int k = j; k <= i - 1; k++)
						{
							V[k][j] -= (f * e[k] + g * d[k]);
						}
						d[j] = V[i - 1][j];
						V[i][j] = 0.0;
					}
				}
				d[i] = h;
			}
			
			// Accumulate transformations.
			
			for (int i = 0; i < n - 1; i++)
			{
				V[n - 1][i] = V[i][i];
				V[i][i] = 1.0;
				double h = d[i + 1];
				if (h != 0.0)
				{
					for (int k = 0; k <= i; k++)
					{
						d[k] = V[k][i + 1] / h;
					}
					for (int j = 0; j <= i; j++)
					{
						double g = 0.0;
						for (int k = 0; k <= i; k++)
						{
							g += V[k][i + 1] * V[k][j];
						}
						for (int k = 0; k <= i; k++)
						{
							V[k][j] -= g * d[k];
						}
					}
				}
				for (int k = 0; k <= i; k++)
				{
					V[k][i + 1] = 0.0;
				}
			}
			for (int j = 0; j < n; j++)
			{
				d[j] = V[n - 1][j];
				V[n - 1][j] = 0.0;
			}
			V[n - 1][n - 1] = 1.0;
			e[0] = 0.0;
		}
		
		// Symmetric tridiagonal QL algorithm.
		
		private void  tql2()
		{
			//  This is derived from the Algol procedures tql2, by
			//  Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
			//  Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
			//  Fortran subroutine in EISPACK.
			
			for (int i = 1; i < n; i++)
			{
				e[i - 1] = e[i];
			}
			e[n - 1] = 0.0;
			
			double f = 0.0;
			double tst1 = 0.0;
			double eps = System.Math.Pow(2.0, - 52.0);
			for (int l = 0; l < n; l++)
			{
				// Find small subdiagonal element
				
				tst1 = System.Math.Max(tst1, System.Math.Abs(d[l]) + System.Math.Abs(e[l]));
				int m = l;
				while (m < n)
				{
					if (System.Math.Abs(e[m]) <= eps * tst1)
					{
						break;
					}
					m++;
				}
				
				// If m == l, d[l] is an eigenvalue,
				// otherwise, iterate.
				
				if (m > l)
				{
					int iter = 0;
					do 
					{
						iter = iter + 1; // (Could check iteration count here.)
						
						// Compute implicit shift
						
						double g = d[l];
						double p = (d[l + 1] - g) / (2.0 * e[l]);
						double r = Maths.Hypot(p, 1.0);
						if (p < 0)
						{
							r = - r;
						}
						d[l] = e[l] / (p + r);
						d[l + 1] = e[l] * (p + r);
						double dl1 = d[l + 1];
						double h = g - d[l];
						for (int i = l + 2; i < n; i++)
						{
							d[i] -= h;
						}
						f = f + h;
						
						// Implicit QL transformation.
						
						p = d[m];
						double c = 1.0;
						double c2 = c;
						double c3 = c;
						double el1 = e[l + 1];
						double s = 0.0;
						double s2 = 0.0;
						for (int i = m - 1; i >= l; i--)
						{
							c3 = c2;
							c2 = c;
							s2 = s;
							g = c * e[i];
							h = c * p;
							r = Maths.Hypot(p, e[i]);
							e[i + 1] = s * r;
							s = e[i] / r;
							c = p / r;
							p = c * d[i] - s * g;
							d[i + 1] = h + s * (c * g + s * d[i]);
							
							// Accumulate transformation.
							
							for (int k = 0; k < n; k++)
							{
								h = V[k][i + 1];
								V[k][i + 1] = s * V[k][i] + c * h;
								V[k][i] = c * V[k][i] - s * h;
							}
						}
						p = (- s) * s2 * c3 * el1 * e[l] / dl1;
						e[l] = s * p;
						d[l] = c * p;
						
						// Check for convergence.
					}
					while (System.Math.Abs(e[l]) > eps * tst1);
				}
				d[l] = d[l] + f;
				e[l] = 0.0;
			}
			
			// Sort eigenvalues and corresponding vectors.
			
			for (int i = 0; i < n - 1; i++)
			{
				int k = i;
				double p = d[i];
				for (int j = i + 1; j < n; j++)
				{
					if (d[j] < p)
					{
						k = j;
						p = d[j];
					}
				}
				if (k != i)
				{
					d[k] = d[i];
					d[i] = p;
					for (int j = 0; j < n; j++)
					{
						p = V[j][i];
						V[j][i] = V[j][k];
						V[j][k] = p;
					}
				}
			}
		}
		
		// Nonsymmetric reduction to Hessenberg form.
		
		private void  orthes()
		{
			//  This is derived from the Algol procedures orthes and ortran,
			//  by Martin and Wilkinson, Handbook for Auto. Comp.,
			//  Vol.ii-Linear Algebra, and the corresponding
			//  Fortran subroutines in EISPACK.
			
			int low = 0;
			int high = n - 1;
			
			for (int m = low + 1; m <= high - 1; m++)
			{
				
				// Scale column.
				
				double scale = 0.0;
				for (int i = m; i <= high; i++)
				{
					scale = scale + System.Math.Abs(H[i][m - 1]);
				}
				if (scale != 0.0)
				{
					
					// Compute Householder transformation.
					
					double h = 0.0;
					for (int i = high; i >= m; i--)
					{
						ort[i] = H[i][m - 1] / scale;
						h += ort[i] * ort[i];
					}
					double g = System.Math.Sqrt(h);
					if (ort[m] > 0)
					{
						g = - g;
					}
					h = h - ort[m] * g;
					ort[m] = ort[m] - g;
					
					// Apply Householder similarity transformation
					// H = (I-u*u'/h)*H*(I-u*u')/h)
					
					for (int j = m; j < n; j++)
					{
						double f = 0.0;
						for (int i = high; i >= m; i--)
						{
							f += ort[i] * H[i][j];
						}
						f = f / h;
						for (int i = m; i <= high; i++)
						{
							H[i][j] -= f * ort[i];
						}
					}
					
					for (int i = 0; i <= high; i++)
					{
						double f = 0.0;
						for (int j = high; j >= m; j--)
						{
							f += ort[j] * H[i][j];
						}
						f = f / h;
						for (int j = m; j <= high; j++)
						{
							H[i][j] -= f * ort[j];
						}
					}
					ort[m] = scale * ort[m];
					H[m][m - 1] = scale * g;
				}
			}
			
			// Accumulate transformations (Algol's ortran).
			
			for (int i = 0; i < n; i++)
			{
				for (int j = 0; j < n; j++)
				{
					V[i][j] = (i == j?1.0:0.0);
				}
			}
			
			for (int m = high - 1; m >= low + 1; m--)
			{
				if (H[m][m - 1] != 0.0)
				{
					for (int i = m + 1; i <= high; i++)
					{
						ort[i] = H[i][m - 1];
					}
					for (int j = m; j <= high; j++)
					{
						double g = 0.0;
						for (int i = m; i <= high; i++)
						{
							g += ort[i] * V[i][j];
						}
						// Double division avoids possible underflow
						g = (g / ort[m]) / H[m][m - 1];
						for (int i = m; i <= high; i++)
						{
							V[i][j] += g * ort[i];
						}
					}
				}
			}
		}
		
		
		// Complex scalar division.
		
		[NonSerialized()]
		private double cdivr, cdivi;

		private void  cdiv(double xr, double xi, double yr, double yi)
		{
			double r, d;
			if (System.Math.Abs(yr) > System.Math.Abs(yi))
			{
				r = yi / yr;
				d = yr + r * yi;
				cdivr = (xr + r * xi) / d;
				cdivi = (xi - r * xr) / d;
			}
			else
			{
				r = yr / yi;
				d = yi + r * yr;
				cdivr = (r * xr + xi) / d;
				cdivi = (r * xi - xr) / d;
			}
		}
		
		
		// Nonsymmetric reduction from Hessenberg to real Schur form.
		
		private void  hqr2()
		{
			//  This is derived from the Algol procedure hqr2,
			//  by Martin and Wilkinson, Handbook for Auto. Comp.,
			//  Vol.ii-Linear Algebra, and the corresponding
			//  Fortran subroutine in EISPACK.
			
			// Initialize
			
			int nn = this.n;
			int n = nn - 1;
			int low = 0;
			int high = nn - 1;
			double eps = System.Math.Pow(2.0, - 52.0);
			double exshift = 0.0;
			double p = 0, q = 0, r = 0, s = 0, z = 0, t, w, x, y;
			
			// Store roots isolated by balanc and compute matrix norm
			
			double norm = 0.0;
			for (int i = 0; i < nn; i++)
			{
				if (i < low | i > high)
				{
					d[i] = H[i][i];
					e[i] = 0.0;
				}
				for (int j = System.Math.Max(i - 1, 0); j < nn; j++)
				{
					norm = norm + System.Math.Abs(H[i][j]);
				}
			}
			
			// Outer loop over eigenvalue index
			
			int iter = 0;
			while (n >= low)
			{
				
				// Look for single small sub-diagonal element
				
				int l = n;
				while (l > low)
				{
					s = System.Math.Abs(H[l - 1][l - 1]) + System.Math.Abs(H[l][l]);
					if (s == 0.0)
					{
						s = norm;
					}
					if (System.Math.Abs(H[l][l - 1]) < eps * s)
					{
						break;
					}
					l--;
				}
				
				// Check for convergence
				// One root found
				
				if (l == n)
				{
					H[n][n] = H[n][n] + exshift;
					d[n] = H[n][n];
					e[n] = 0.0;
					n--;
					iter = 0;
					
					// Two roots found
				}
				else if (l == n - 1)
				{
					w = H[n][n - 1] * H[n - 1][n];
					p = (H[n - 1][n - 1] - H[n][n]) / 2.0;
					q = p * p + w;
					z = System.Math.Sqrt(System.Math.Abs(q));
					H[n][n] = H[n][n] + exshift;
					H[n - 1][n - 1] = H[n - 1][n - 1] + exshift;
					x = H[n][n];
					
					// Real pair
					
					if (q >= 0)
					{
						if (p >= 0)
						{
							z = p + z;
						}
						else
						{
							z = p - z;
						}
						d[n - 1] = x + z;
						d[n] = d[n - 1];
						if (z != 0.0)
						{
							d[n] = x - w / z;
						}
						e[n - 1] = 0.0;
						e[n] = 0.0;
						x = H[n][n - 1];
						s = System.Math.Abs(x) + System.Math.Abs(z);
						p = x / s;
						q = z / s;
						r = System.Math.Sqrt(p * p + q * q);
						p = p / r;
						q = q / r;
						
						// Row modification
						
						for (int j = n - 1; j < nn; j++)
						{
							z = H[n - 1][j];
							H[n - 1][j] = q * z + p * H[n][j];
							H[n][j] = q * H[n][j] - p * z;
						}
						
						// Column modification
						
						for (int i = 0; i <= n; i++)
						{
							z = H[i][n - 1];
							H[i][n - 1] = q * z + p * H[i][n];
							H[i][n] = q * H[i][n] - p * z;
						}
						
						// Accumulate transformations
						
						for (int i = low; i <= high; i++)
						{
							z = V[i][n - 1];
							V[i][n - 1] = q * z + p * V[i][n];
							V[i][n] = q * V[i][n] - p * z;
						}
						
						// Complex pair
					}
					else
					{
						d[n - 1] = x + p;
						d[n] = x + p;
						e[n - 1] = z;
						e[n] = - z;
					}
					n = n - 2;
					iter = 0;
					
					// No convergence yet
				}
				else
				{
					
					// Form shift
					
					x = H[n][n];
					y = 0.0;
					w = 0.0;
					if (l < n)
					{
						y = H[n - 1][n - 1];
						w = H[n][n - 1] * H[n - 1][n];
					}
					
					// Wilkinson's original ad hoc shift
					
					if (iter == 10)
					{
						exshift += x;
						for (int i = low; i <= n; i++)
						{
							H[i][i] -= x;
						}
						s = System.Math.Abs(H[n][n - 1]) + System.Math.Abs(H[n - 1][n - 2]);
						x = y = 0.75 * s;
						w = (- 0.4375) * s * s;
					}
					
					// MATLAB's new ad hoc shift
					
					if (iter == 30)
					{
						s = (y - x) / 2.0;
						s = s * s + w;
						if (s > 0)
						{
							s = System.Math.Sqrt(s);
							if (y < x)
							{
								s = - s;
							}
							s = x - w / ((y - x) / 2.0 + s);
							for (int i = low; i <= n; i++)
							{
								H[i][i] -= s;
							}
							exshift += s;
							x = y = w = 0.964;
						}
					}
					
					iter = iter + 1; // (Could check iteration count here.)
					
					// Look for two consecutive small sub-diagonal elements
					
					int m = n - 2;
					while (m >= l)
					{
						z = H[m][m];
						r = x - z;
						s = y - z;
						p = (r * s - w) / H[m + 1][m] + H[m][m + 1];
						q = H[m + 1][m + 1] - z - r - s;
						r = H[m + 2][m + 1];
						s = System.Math.Abs(p) + System.Math.Abs(q) + System.Math.Abs(r);
						p = p / s;
						q = q / s;
						r = r / s;
						if (m == l)
						{
							break;
						}
						if (System.Math.Abs(H[m][m - 1]) * (System.Math.Abs(q) + System.Math.Abs(r)) < eps * (System.Math.Abs(p) * (System.Math.Abs(H[m - 1][m - 1]) + System.Math.Abs(z) + System.Math.Abs(H[m + 1][m + 1]))))
						{
							break;
						}
						m--;
					}
					
					for (int i = m + 2; i <= n; i++)
					{
						H[i][i - 2] = 0.0;
						if (i > m + 2)
						{
							H[i][i - 3] = 0.0;
						}
					}
					
					// Double QR step involving rows l:n and columns m:n
					
					for (int k = m; k <= n - 1; k++)
					{
						bool notlast = (k != n - 1);
						if (k != m)
						{
							p = H[k][k - 1];
							q = H[k + 1][k - 1];
							r = (notlast?H[k + 2][k - 1]:0.0);
							x = System.Math.Abs(p) + System.Math.Abs(q) + System.Math.Abs(r);
							if (x != 0.0)
							{
								p = p / x;
								q = q / x;
								r = r / x;
							}
						}
						if (x == 0.0)
						{
							break;
						}
						s = System.Math.Sqrt(p * p + q * q + r * r);
						if (p < 0)
						{
							s = - s;
						}
						if (s != 0)
						{
							if (k != m)
							{
								H[k][k - 1] = (- s) * x;
							}
							else if (l != m)
							{
								H[k][k - 1] = - H[k][k - 1];
							}
							p = p + s;
							x = p / s;
							y = q / s;
							z = r / s;
							q = q / p;
							r = r / p;
							
							// Row modification
							
							for (int j = k; j < nn; j++)
							{
								p = H[k][j] + q * H[k + 1][j];
								if (notlast)
								{
									p = p + r * H[k + 2][j];
									H[k + 2][j] = H[k + 2][j] - p * z;
								}
								H[k][j] = H[k][j] - p * x;
								H[k + 1][j] = H[k + 1][j] - p * y;
							}
							
							// Column modification
							
							for (int i = 0; i <= System.Math.Min(n, k + 3); i++)
							{
								p = x * H[i][k] + y * H[i][k + 1];
								if (notlast)
								{
									p = p + z * H[i][k + 2];
									H[i][k + 2] = H[i][k + 2] - p * r;
								}
								H[i][k] = H[i][k] - p;
								H[i][k + 1] = H[i][k + 1] - p * q;
							}
							
							// Accumulate transformations
							
							for (int i = low; i <= high; i++)
							{
								p = x * V[i][k] + y * V[i][k + 1];
								if (notlast)
								{
									p = p + z * V[i][k + 2];
									V[i][k + 2] = V[i][k + 2] - p * r;
								}
								V[i][k] = V[i][k] - p;
								V[i][k + 1] = V[i][k + 1] - p * q;
							}
						} // (s != 0)
					} // k loop
				} // check convergence
			} // while (n >= low)
			
			// Backsubstitute to find vectors of upper triangular form
			
			if (norm == 0.0)
			{
				return ;
			}
			
			for (n = nn - 1; n >= 0; n--)
			{
				p = d[n];
				q = e[n];
				
				// Real vector
				
				if (q == 0)
				{
					int l = n;
					H[n][n] = 1.0;
					for (int i = n - 1; i >= 0; i--)
					{
						w = H[i][i] - p;
						r = 0.0;
						for (int j = l; j <= n; j++)
						{
							r = r + H[i][j] * H[j][n];
						}
						if (e[i] < 0.0)
						{
							z = w;
							s = r;
						}
						else
						{
							l = i;
							if (e[i] == 0.0)
							{
								if (w != 0.0)
								{
									H[i][n] = (- r) / w;
								}
								else
								{
									H[i][n] = (- r) / (eps * norm);
								}
								
								// Solve real equations
							}
							else
							{
								x = H[i][i + 1];
								y = H[i + 1][i];
								q = (d[i] - p) * (d[i] - p) + e[i] * e[i];
								t = (x * s - z * r) / q;
								H[i][n] = t;
								if (System.Math.Abs(x) > System.Math.Abs(z))
								{
									H[i + 1][n] = (- r - w * t) / x;
								}
								else
								{
									H[i + 1][n] = (- s - y * t) / z;
								}
							}
							
							// Overflow control
							
							t = System.Math.Abs(H[i][n]);
							if ((eps * t) * t > 1)
							{
								for (int j = i; j <= n; j++)
								{
									H[j][n] = H[j][n] / t;
								}
							}
						}
					}
					
					// Complex vector
				}
				else if (q < 0)
				{
					int l = n - 1;
					
					// Last vector component imaginary so matrix is triangular
					
					if (System.Math.Abs(H[n][n - 1]) > System.Math.Abs(H[n - 1][n]))
					{
						H[n - 1][n - 1] = q / H[n][n - 1];
						H[n - 1][n] = (- (H[n][n] - p)) / H[n][n - 1];
					}
					else
					{
						cdiv(0.0, - H[n - 1][n], H[n - 1][n - 1] - p, q);
						H[n - 1][n - 1] = cdivr;
						H[n - 1][n] = cdivi;
					}
					H[n][n - 1] = 0.0;
					H[n][n] = 1.0;
					for (int i = n - 2; i >= 0; i--)
					{
						double ra, sa, vr, vi;
						ra = 0.0;
						sa = 0.0;
						for (int j = l; j <= n; j++)
						{
							ra = ra + H[i][j] * H[j][n - 1];
							sa = sa + H[i][j] * H[j][n];
						}
						w = H[i][i] - p;
						
						if (e[i] < 0.0)
						{
							z = w;
							r = ra;
							s = sa;
						}
						else
						{
							l = i;
							if (e[i] == 0)
							{
								cdiv(- ra, - sa, w, q);
								H[i][n - 1] = cdivr;
								H[i][n] = cdivi;
							}
							else
							{
								
								// Solve complex equations
								
								x = H[i][i + 1];
								y = H[i + 1][i];
								vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q;
								vi = (d[i] - p) * 2.0 * q;
								if (vr == 0.0 & vi == 0.0)
								{
									vr = eps * norm * (System.Math.Abs(w) + System.Math.Abs(q) + System.Math.Abs(x) + System.Math.Abs(y) + System.Math.Abs(z));
								}
								cdiv(x * r - z * ra + q * sa, x * s - z * sa - q * ra, vr, vi);
								H[i][n - 1] = cdivr;
								H[i][n] = cdivi;
								if (System.Math.Abs(x) > (System.Math.Abs(z) + System.Math.Abs(q)))
								{
									H[i + 1][n - 1] = (- ra - w * H[i][n - 1] + q * H[i][n]) / x;
									H[i + 1][n] = (- sa - w * H[i][n] - q * H[i][n - 1]) / x;
								}
								else
								{
									cdiv(- r - y * H[i][n - 1], - s - y * H[i][n], z, q);
									H[i + 1][n - 1] = cdivr;
									H[i + 1][n] = cdivi;
								}
							}
							
							// Overflow control
							
							t = System.Math.Max(System.Math.Abs(H[i][n - 1]), System.Math.Abs(H[i][n]));
							if ((eps * t) * t > 1)
							{
								for (int j = i; j <= n; j++)
								{
									H[j][n - 1] = H[j][n - 1] / t;
									H[j][n] = H[j][n] / t;
								}
							}
						}
					}
				}
			}
			
			// Vectors of isolated roots
			
			for (int i = 0; i < nn; i++)
			{
				if (i < low | i > high)
				{
					for (int j = i; j < nn; j++)
					{
						V[i][j] = H[i][j];
					}
				}
			}
			
			// Back transformation to get eigenvectors of original matrix
			
			for (int j = nn - 1; j >= low; j--)
			{
				for (int i = low; i <= high; i++)
				{
					z = 0.0;
					for (int k = low; k <= System.Math.Min(j, high); k++)
					{
						z = z + V[i][k] * H[k][j];
					}
					V[i][j] = z;
				}
			}
		}

		#endregion //  Private Methods
		
		
		#region Constructor
		
		/// <summary>Check for symmetry, then construct the eigenvalue decomposition</summary>
		/// <param name="Arg">   Square matrix
		/// </param>
		/// <returns>     Structure to access D and V.
		/// </returns>
		
		public EigenvalueDecomposition(GeneralMatrix Arg)
		{
			double[][] A = Arg.Array;
			n = Arg.ColumnDimension;
			V = new double[n][];
			for (int i = 0; i < n; i++)
			{
				V[i] = new double[n];
			}
			d = new double[n];
			e = new double[n];
			
			issymmetric = true;
			for (int j = 0; (j < n) & issymmetric; j++)
			{
				for (int i = 0; (i < n) & issymmetric; i++)
				{
					issymmetric = (A[i][j] == A[j][i]);
				}
			}
			
			if (issymmetric)
			{
				for (int i = 0; i < n; i++)
				{
					for (int j = 0; j < n; j++)
					{
						V[i][j] = A[i][j];
					}
				}
				
				// Tridiagonalize.
				tred2();
				
				// Diagonalize.
				tql2();
			}
			else
			{
				H = new double[n][];
				for (int i2 = 0; i2 < n; i2++)
				{
					H[i2] = new double[n];
				}
				ort = new double[n];
				
				for (int j = 0; j < n; j++)
				{
					for (int i = 0; i < n; i++)
					{
						H[i][j] = A[i][j];
					}
				}
				
				// Reduce to Hessenberg form.
				orthes();
				
				// Reduce Hessenberg to real Schur form.
				hqr2();
			}
		}

		#endregion //  Constructor
		
		#region Public Properties
		/// <summary>Return the real parts of the eigenvalues</summary>
		/// <returns>     real(diag(D))
		/// </returns>
		virtual public double[] RealEigenvalues
		{
			get
			{
				return d;
			}
		}
		/// <summary>Return the imaginary parts of the eigenvalues</summary>
		/// <returns>     imag(diag(D))
		/// </returns>
		virtual public double[] ImagEigenvalues
		{
			get
			{
				return e;
			}
		}
		/// <summary>Return the block diagonal eigenvalue matrix</summary>
		/// <returns>     D
		/// </returns>
		virtual public GeneralMatrix D
		{
			get
			{
				GeneralMatrix X = new GeneralMatrix(n, n);
				double[][] D = X.Array;
				for (int i = 0; i < n; i++)
				{
					for (int j = 0; j < n; j++)
					{
						D[i][j] = 0.0;
					}
					D[i][i] = d[i];
					if (e[i] > 0)
					{
						D[i][i + 1] = e[i];
					}
					else if (e[i] < 0)
					{
						D[i][i - 1] = e[i];
					}
				}
				return X;
			}
		}
		#endregion //  Public Properties
		
		#region Public Methods
		
		/// <summary>Return the eigenvector matrix</summary>
		/// <returns>     V
		/// </returns>
		
		public virtual GeneralMatrix GetV()
		{
			return new GeneralMatrix(V, n, n);
		}
		#endregion //  Public Methods

		// A method called when serializing this class.
		void ISerializable.GetObjectData(SerializationInfo info, StreamingContext context) 
		{
		}
	}
}